Excel in math, science, and engineering

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\[\lim_{x \to \infty} \frac{2\sqrt{3} \sin \left(\frac{\sqrt{3}}{x}\right)-3 \sin \left(\frac{2}{x}\right) }{\ln \left( \sqrt{2} \sin \left( \frac{\pi x + 4}{4x}\right) \right)\left(1-\cos(\cot^{-1}x)\right)} = \, ? \]

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